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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
1. Black-Scholes formula.
2. Change of variables for Kolmogorov equation.
3. Mean reverting equation.
4. Affine SDE.
5. Heston equations.
6. Displaced Heston equations.
7. Stochastic volatility.
8. Markovian projection.
A. Markovian projection on displaced diffusion.
B. Markovian projection on Heston model.
9. Hamilton-Jacobi Equations.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Markovian projection on Heston model.


ollowing sections ( Markovian projection ) and ( Markovian projection on displaced diffusion ) we are developing a generic recipe for approximation of the process $X_{t}$ given by the SDE

MATH (MarkPr TargetEquation)
with some adapted diffusion process $\omega_{t}$ . We will be using Heston-type processes as the class of approximating processes.

The reference for this section is [Antonov2007] .

In line with the section ( Heston_equation_section ) we seek approximation in the class of the stochastic processes given by the equations

MATH (Heston approximation)
for variety of deterministic functions MATH . The $dW_{k,t},~k=1,2$ are standard Brownian motions. The $\Delta$ refers to the difference MATH . The analytical tractability of the equations ( Heston approximation ) was considered in the sections ( Heston equation ) and ( Displaced diffusion ).

Note that we may write the equation ( MarkPr TargetEquation ) in the equivalent form

MATH (MarkPr TargetEquations 2)
for some diffusion processes $V_{t}$ , $a_{t}$ , $b_{1,t}$ and $b_{2,t}.$ To see that the equations ( MarkPr TargetEquations 2 ) are equivalent to ( MarkPr TargetEquation ) observe that we may introduce the process MATH and claim that it a diffusion because $\omega_{t}$ is assumed to be a diffusion. Consequently, we introduce the diffusion process MATH The process $V_{t}$ have correlated and uncorrelated components of the diffusion term, hence, we have the sum MATH for some processes $b_{k,t}$ , $k=1,2$ . Finally, $V_{t}$ is positive, hence the multiplication by $\sqrt{V_{t}}$ does not restrict the generality. The motivation behind such transformation is the aim to present the target process $X_{t}$ in the form similar to the form of the approximating process $Y_{t}$ .

We will be applying the Gyongy's result ( Multidimensional Gyongy lemma ), hence, we put the target and approximating processes in the matrix form. We seek to approximate MATH with MATH given by the SDEs MATH MATH Note, that according to the lemma ( Multidimensional Gyongy lemma ) a general 2-dimensional process MATH may be replicated by the local volatility process MATH with the functions MATH , MATH given by the relationships MATH We note that in our setting MATH and MATH . Following motivations of the previous section ( Markovian projection section ) the functions MATH , MATH minimize the following functionals: MATH We substitute the expressions for $a,b,\alpha,\beta$ : MATH and simplify to MATH

The rest of the calculation is a straightforward extension of the previous section ( Markovian projection on displaced diffusion ). We calculate the variations of the functional $F_{1}$ MATH MATH We minimize the functional $F_{3}$ : MATH We calculate variations of the functional $F_{4}$ : MATH MATH We summarize the results of minimization: MATH The functions MATH are defined by (a) and (b). The absolute values MATH is defined by (c). The MATH and MATH are defined by (d) and (e). The expectations are calculated by techniques outlined in the previous two sections.





Notation. Index. Contents.


















Copyright 2007